Conclusions

6.5 Conclusions

When following the non linear evolution of strange mode instabilities in the envelopes of massive stars, shock fronts were observed to be captured in the H-ionisation zone some pulsation periods after reaching the non linear regime. This effect is not observed in models of very hot envelopes (such as the massive star model investigated by Glatzel et al. (1999)), due to hydrogen being ionised completely. The shocks trapped in the H-ionisation zone perform high frequency oscillations (associated with the sound travel times across the shock zone) confined to its very vicinity, whereas the remaining parts of the envelope vary on the dynamical timescale of the primary, strange mode instability.

By performing an appropriate linear stability analysis the high frequency oscillations were shown to be due to a physical instability, rather than being a numerical artifact.

An analytical model for the secondary, shock zone instabilities has been constructed.

As a result, high values of∇ were found to be responsible for instability. Contrary to the common stratification (convective, Rayleigh-Taylor) instabilities driven by buoyancy forces and thus associated with (non radial) gravity modes, however, the instabilities found here are associated with spherically symmetric acoustic waves. An extension of the stability analysis to non radial perturbations would be instructive, since we expect the acoustic instabilities identified here - similar to strange mode instabilities (see Glatzel &

Mehren 1996) - not to be restricted to spherical geometry. Such an investigation would also reveal buoyancy driven instabilities, which we believe not to be relevant for the following reasons: Their typical timescale is much longer than that of the acoustic insta-bilities, which will therefore dominate the dynamics. Moreover, in addition to gravity, the acceleration due to the shocks velocity field has to be taken into account and is likely to stabilise the stratification with respect to convective instabilities. With respect to the aim of this paper to identify the secondary, shock oscillations and their origin, a non radial analysis is beyond the scope of the present investigation and will be the subject of a forthcoming publication.

Since the oscillations are a physical phenomenon rather than a numerical artifact -they should not be damped by increasing the artificial viscosity as one would neglect a physical process whose influence on the long term behaviour of the system cannot yet be predicted. On the other hand, following the shock oscillations by numerical simulation for more than a few dynamical timescales is not feasible due to the small timesteps necessary to resolve them. The confinement of the oscillations to the very vicinity of hydrogen ionisation, however, indicates a solution of the problem by means of domain decomposition: The stellar envelope is decomposed into three domains: below, around and above the shock. Only the narrow shock region needs high time resolution, the inner and outer zones merely require the dynamical timescale to be resolved. The development of a code following this strategy is in progress.

Even if the appearance of the shock oscillations has so far prevented us from perform-ing simulations in excess of several dynamical timescales, the velocity amplitudes reach a significant fraction of the escape velocity. This indicates that pulsationally driven mass loss may be found in appropriate simulations. Whether the new code will allow for the

CHAPTER 6. CAPTURED SHOCKS

corresponding long term simulations and thus possibly for the determination of mass loss rates, remains to be seen. Preliminary results will be published in a forthcoming paper.

Acknowledgements

We thank Professor K.J. Fricke for encouragement and support. Financial support by the Graduiertenkolleg “Strömungsinstabilitäten und Turbulenz” (MG) and by the DFG under grant WA 633 12-1 (SC) is gratefully acknowledged. The numerical computations have been carried out using the facilities of the GWDG at Göttingen.

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BIBLIOGRAPHY

7. Domain decomposition

Simulation of stellar instabilities with vastly different time-scales using domain decomposition

M. Grott

, S. Chernigovski

and W. Glatzel

2Universitäts-Sternwarte Göttingen, Geismarlandstr .11, 37083 Göttingen, Germany

3Institut für Analysis und Numerik, Universität Magdeburg, Universitätsplatz 2, 39106 Magdeburg, Germany

Abstract

Strange mode instabilities in the envelopes of massive stars lead to shock waves, which can oscillate on a much shorter timescale than that associated with the primary instability. The phenomenon is studied by direct numerical simulation using a, with respect to time, implicit Lagrangian scheme, which allows for the variation by several orders of magnitude of the dependent variables. The timestep for the simulation of the system is reduced appreciably by the shock oscillations and prevents its long term study. A pro-cedure based on domain decomposition is proposed to surmount the difficulty of vastly different timescales in various regions of the stellar envelope and thus to enable the desired long term sim-ulations. Criteria for domain decomposition are derived and the proper treatment of the resulting inner boundaries is discussed.

Tests of the approach are presented and its viability is demon-strated by application to a model for the star P Cygni. In this in-vestigation primarily the feasibility of domain decomposition for the problem considered is studied. We intend to use the results as the basis of an extension to two dimensional simulations.

Key words: hydrodynamics - instabilities - shock waves - stars:

oscillations - stars: variables: other - stars: individual: P Cygni.

1This paper has been accepted for publication by MNRAS (Grott et al. 2003b)

CHAPTER 7. DOMAIN DECOMPOSITION

7.1 Introduction

Sufficiently luminous objects, such as massive stars, are known to suffer from strange mode instabilities with growth rates in the dynamical range (Kiriakidis, Fricke & Glatzel 1993, Glatzel & Kiriakidis 1993). The boundary of the domain in the Hertzsprung-Russel diagram (HRD) above which all stellar models are unstable - irrespective of their metallicity -, coincides with the observed Humphreys-Davidson (HD) limit (Humphreys

& Davidson 1979). Moreover, the range of unstable models covers the stellar parameters for which the LBV (luminous blue variable) phenomenon is observed (for a review see Humphreys & Davidson 1994).

The high growth rates of the instabilities indicate a connection to the observed mass loss of the corresponding objects. To verify this suggestion, simulations of their evolu-tion into the non linear regime have been performed. In fact, for selected models Glatzel et al. (1999) found the velocity amplitude to exceed the escape velocity (see, however, Dorfi & Gautschy (2000)).

To identify a possible connection between non linear pulsations and outbursts in lu-minous blue variables Grott, Glatzel & Chernigovski (2003) have studied the evolution of an initial model located in the HRD well above the HD limit. In this study, the shocks formed in the non linear regime are captured by the H-ionisation zone after a few pul-sation periods. These captured shocks start to oscillate rapidly with periods of the order of the sound travel time across the H-ionisation zone, while its mean position changes on the dynamical timescale of the primary, strange mode instability. Grott et al. (2003) have shown, that this shock front oscillation is of physical origin and therefore must not be disregarded. In particular, the phenomenon should not be eliminated by increasing the artificial viscosity. We note that the representation of the phenomenon requires the correct treatment of extreme gradients of the dependent variables, implying their vari-ation by several orders of magnitude. It is achieved by use of a, with respect to time implicit Lagrangian scheme.

The rapid shock oscillations, which are confined to a narrow region in the vicinity of the shock front, require an inhibitively small timestep and thus prevent long term sim-ulations. In the present paper we propose an approach based on domain decomposition to surmount the difficulty of vastly different timescales in various regions of the stellar envelope and thus to enable the desired long term simulations. In this procedure the various domains within the envelope are to be treated separately and according to their intrinsic timescales. We expect this decomposition to speed up the calculations consid-erably. An even higher speedup will be achieved when applying domain decomposition to two dimensional simulations. In this sense, the present investigation may be regarded as a preliminary study for decomposition in two dimensions.

The basic equations and assumptions are introduced in Section 7.2. The domain de-composition approach is discussed in detail in Section 7.3, including a derivation of criteria for domain decomposition and an investigation of the proper treatment of the resulting inner boundaries. Moreover, tests of the approach are presented there. Its via-bility is demonstrated by application to a model of the star P Cygni in section 7.4. Our